Widdekind

Not new (in essence) ??

As seen on this site, the Oberon moon of Uranus shows a 6 km peak, protruding from the (lower-left) limb of its Voyager 2 photograph: This image of Oberon shows several large impact craters towards the center of the picture. Many of the crater floors are covered by an unknown dark material. On the bottom-left limb, a high mountain rises 6 kilometers (4 miles) above its surroundings. Bright rays similar to those seen on Jupiter's moon Callisto, can be found on Oberon's surface. (Copyright Calvin J. Hamilton) CONCLUSION (??): This 6 km peak, is actually the central peak (140° east, 25° south), of a crater, which is roughly as wide, as the Mommur Chasma is long — to wit, roughly 500-550 km — and to which there could, conceivably, be some kind of connection (cf. Mercury's Weird Terrain is antipodal to its Caloris Basin).

Merely as a mathematical question, imagine replacing the Dirac Delta Function, in the above formula, with the Wave Function of some stable (bound) state (in one of the atoms in the detector), into which you "wanted" an incoming electron to collapse. For example, something along the lines of the standard hydrogen atom WFs. Mathematically speaking, wouldn't that "make", "force", or "hammer" the electron's own WF into the desired bound state ? For example, wouldn't the imaginary potential ¡ ( 2 ¥1s - 1 ) "grow" the electron WF into ¥1s (~ er / r) ?? (If you multiplied that unit-less formula, by the Planck Energy, wouldn't it "grow" the electron WF, into the desired bound state, in a time of order the Planck Time ??) Of course, the standard Overlap Integral < ¥e | ¥1s > would also seem to yield a probability, of the electron (in ¥e) appearing in the desired bound state (in ¥1s). But, how could you turn that probability into a percentage chance of "collapse" into said bound state ??

Recent theories, regarding the origin of Saturn's rings, link them to that planet's icy inner moons — "the tiny particles that form the rings today... [are] pieces of moon" (Ron Miller. Saturn, pg. 32). However, evidence indicates that the rings are relatively young: There are many reasons that the rings are considered relatively young. One of these is how clean the ring particles are. The rings are very bright be/c the ice particles they are made of have not yet had time to be covered with dark dust. Another reason is that the gravitational effects of all of Saturn's moons — the same forces that create the thousands of large and small gaps in the rings — make the rings unstable. They look the way they do now only b/c the moons haven't had enough time yet to disrupt them. A few million years from now, however, the rings will start to fall in toward Saturn, ad the solar system will lose one of its greatest natural wonders (Ron Miller, ibid.). Now, it is also suggested, that some of Saturn's icy inner moons may mount geysers, powered by tidal heating: Some astronomers have speculated that Enceladus may have geysers, powered by the heat generated beneath the surface of the moon due to tidal flexing caused by nearby Saturn (Ron Miller, ibid., pg. 64). CONCLUSION (??): Could the constant tidal heating, of the icy inner moons by Saturn, create constant geysers, whose high speed streams escape those moons' weak gravities, and constantly replenish the rings (as fast as the fall inward) ?? If so, then, in-so-far as the icy inner moons' supplies of volatiles has decreased over the eons, Saturn's rings may have been bigger & brighter in the dim & distant past. Ice Fracture on Enceladus — Voyager photos in 1979 AD showed that Enceladus has more pronounced fracturing and resurfacing than other Saturn moons. Here we look along a geologically young fracture toward Saturn, which subtends 29°. Angular width: 45°. (Out of the Cradle, pp. 158-159.) Eruption on Enceladus — The brightness of Enceladus' ice, its sparsely cratered, fractured plains, and the nearby E Ring of ice crystals, [all] suggest that Enceladus may be geologically active. Here an eruption of water & vapor blows a fresh supply of ice crystals off Enceladus and into the E Ring. (Out of the Cradle, pg. 163.) Eclipse of the sun seen from Tethys — Reddened by shining through Saturn's atmosphere, the Sun casts a sunset glow across Tethys' [cratered] ice fields. The puzzling moon Enceladus, covered by the brightest ice in the solar system, is at upper left. The Sun backlights the mysterious E Ring, a fuzzy extension of Saturn's main ring system. The E Ring consists of microscopic ice crystals, concentrated along Enceladus' orbit, which may be evidence of fresh ice-volcano eruptions blowing material off Enceladus. Angular width of wide-angle view: 70°. (Out of the Cradle, pp. 160-161) Our legacy on Mimas — Seeking clues as to whether icy Mimas was completely disrupted & reassembled one or more times, by the intense cratering it has undergone due to meteorite impact, explorers have left their mark on this innermost of Saturn's sizeable moons. Saturn, subtending 39°, covers much of the sky. (Out of the Cradle, pg. 162.) Merged post follows: Consecutive posts merged Uranus is known to have 21 moons and 9 "thin wispy" rings; Neptune is known to have 8 moons and 4 "very narrow" rings (Ron Miller. Uranus & Neptune, pp. 35,43,52,56). It is also likely that the ring particles are "smashed moon... debris [which] would have provided material for Neptune's rings" (ibid., pp. 46). And, tide-generated geysers are known to exist on Neptune's moon Triton (ibid., pg. 51) CONCLUSION (??): These facts are completely consistent w/ claims, that tide-generated geysers on moons, spew out volatile materials, which make up the parent planets' rings. Indeed, fore the aforesaid Ice Giants, the number of rings (8,4) correlates closely with the number of moons (21,8), in a ratio quite close to 2:1. Indeed also, Saturn sports some 62 moons, many more than either Ice Giant, again completely consistent w/ these claims*. Moreover, since said Ice Giants are much less massive than Saturn, they generate weaker tidal forces, which would generate weaker geysers, which would explain the Ice Giants' far fainter ring systems. Moreover still, these dramatically reduced ring replenishment rates easily explain the "very dark" appearance of the Ice Giants' rings, in stark contrast to Saturn — "[the rings] are very dark, resembling powdered coal more than the chunks of [fresh] ice that make up Saturn's rings" — since the Ice Giants' rings have had more time to sweep up dust than those of Saturn. * Jupiter boasts 63 moons, but only 8 are regular satellites (which formed along with their parent planet), whilst "Jupiter's other 55 moons are irregular satellites, whose prograde and retrograde orbits are much farther from Jupiter and have high inclinations and eccentricities [and] were likely captured by Jupiter from solar orbits". Somewhat similarly, Saturn sports 24 regular moons, whilst "the remaining 38, all small save one, are irregular satellites, whose orbits are much farther from Saturn, have high inclinations, and are mixed between prograde and retrograde [and which] were likely captured minor planets, or debris from the breakup of such bodies after they were captured, creating collisional families". Thus, Saturn sports three-times more moons, on regular interior orbits near its rings, than Jupiter, yet again completely consistent w/ these claims. Moreover, since Jupiter is much more massive, and generates significantly stronger tides, its inner moons may be "dynamically older", having finished venting off their volatiles long ago, leaving Jupiter's rings today in something of a state of "slow decay", well past their prime. PREDICTIONS: Geysers indicate that moons are molten in the middle. Such is associated w/ volcanic and other "planeto-thermal" activities which rapidly resurface those moons. Thus, rings, being linked to geysers, ought to be linked to young surfaces on the moons near those rings. Neptune's moon Triton orbits retrograde, at high inclination, well away from its parent planet's rings. Yet, geysers still spew from its surface. So, unless Triton's exceptional size & gravity re-capture all of that ejecta, Neptune should show some sort of ring associated w/ Triton's orbit. In-so-far as "young moon surfaces" are associated w/ ring systems, then the most spectacular rings around ought to be in young, freshly formed, star systems. To wit, "somewhere out there", is a super-Saturnianly-spectacular ultra-ring system, waiting for Astronomers & Exoplanetologists to take its picture. Merged post follows: Consecutive posts merged Jupiter's rings are "dark", resembling dust, and are "closer to Jupiter than its moons" (Ron Miller. Jupiter, pg. 48). This is completely consistent w/ Jupiter's rings being "dynamically older" than those of Saturn (and even the Ice Giants), since Jupiter's super-tides long ago boiled off the volatiles from Jupiter's inner-most moons, whose once-mighty (?) ice geysers now stand silent & extinct. Thus, Jupiter's faint dusky rings are already past their prime, in a state of inexorable decline, sliding ever planet-ward. Indeed: Jupiter has a ring of microscopic, dark-colored particles extending inward from the region of four small moons that line inside Io's orbit. The largest of these moons, Amalthea, is a potato-shaped lump about 155 x 200 km (97 x 125 mi). It has a reddish color, possibly derived from sulfur materials knocked off Io by meteorites or energetic atomic particles. The other moons are much smaller: 35 to 75 km (22 to 47 mi) across. Tiny particles knocked off these moons, especially the one on the very edge of the ring [Metis], would spiral in toward Jupiter, and may be the source of the ring. In this view, the ring with its inward flow of particles would be like a [inwardly spiraling] river — always there, but containing different material at each moment... from the ring's fuzzy inner edge, to its sharp out edge near the moonlets' orbits (W.K. Hartmann, R. Miller, P. Lee. Out of the Cradle, pg. 152). Again, this agrees with our previous picture of the process. In Jupiter's Ring — We float above a small moonlet (bottom) at the outer edge of Jupiter's thin, enigmatic ring of dark, stony particles. Looking toward Jupiter, we see the surreal cloud pattern of Jupiter's Red Spot (top) and the gray ring stretching away thousands of miles into the distance (bottom half), creating its own 'horizon'. The ring is thin enough that we can dimly see through it. (Out of the Cradle, pg. 152.) Merged post follows: Consecutive posts merged The early Earth possessed a ring, of dust & debris, in the wake of the Moon-forming impact, roughly 4.5 Gya: The collision probably took place when the Earth was only about half formed. The impact threw a ring of very hot debris into orbit around the young Earth. The Moon probably formed by accretion very quickly from this debris, perhaps in fewer than 10 years... The newly formed Moon began to gradually spiral farther & farther away from Earth. One hundred million years after its creation — 4.4 billion years ago — the Moon was already half-way to its present distance (Ron Miller. Earth & Moon, pp. 17-19). This strongly suggests, that any process which deposits dust & debris, into orbit about a particular planet, causes the creation of a ring. Likewise, if that process is, itself, ongoing, then it can maintain that ring. Note that ring systems seem inherently unstable, falling ever inwards. For, so much of that dust & debris, in Earth's brief ring, fell into Earth's upper atmosphere, that it darkened Earth's skies for roughly 300 million years: 4.4 billion years ago... the sky would have been dark because of the high-altitude layer of dust [infalling from the Earth's dusty debris ring, as well as relic dust from the Sun's planetary disk]... The sky began to brighten about 4.2 billion years ago, even though asteroids still fell (ibid., pg. 24). Earth's skies may not have cleared completely until after 3.8 Gya (ibid., pp. 28-29).

In Classical physics, particles are pure points, which form World-lines through Spacetime. Conversely, Quantum physics pictures particles as Wave Functions which are extended in space. Are Wave Functions extended in time, too ?

What causes WF collapse ? In the Schrodinger Wave Equation, wouldn't an imaginary potential cause, at least on its own, WFs to exponentially grow (positive imaginary, same sign as time derivative) or exponentially shrink (negative imaginary, opposite sign from same). Could that, conceivably, have something to do with the "collapse of WFs" during observation & measurement ? For example, using the 3D Dirac Delta Function ∆(x), the imaginary PE field Î(2∆(x)-1) could conceivably cause a wave function to exponentially grow at x, whilst exponentially shrinking away everywhere else (?).

Further Questions: Back when all matter was formless energy, in the pre-Big-Bang singularity, did all matter-energy become mutually QE'd, and did that QE survive the fiery birth of the Universe ? If antimatter is "negative energy matter", such that it's appearance is always heralded by powerful explosions of energy... then why doesn't "normal" matter spontaneously decay into antimatter, following the energy gradient, as in (free) neutron decay ? Merged post follows: Consecutive posts merged Wow — so, "consciousness cleaves reality, from possibility" ? What happens to the "other reality", is it "pruned" from possibility, into ignominy ? Where would the matter-energy come from, to keep creating copious quantities of Cosmoses, every time a consciousness, somewhere & when, came to a new conclusion ??

According to Wikipedia, to explain the non-locality of Quantum Entanglement, one must invoke velocities of at least ten thousand times the speed-of-light (104c). In my "retrocausal diagram", the information would appear to flow, at these hyper-luminal speeds, "down" the world-line of the one entangled particle, "to" the Quantum Entanglement Event in Spacetime, and then "up" from there along the world-line of the other entangled particle, to the "present", where that information would induce the appropriate quantum behavior in the paired particle. Merged post follows: Consecutive posts merged ANOTHER QUESTION — Breaking Entanglement ?? According to Wikipedia, Retrocausality interprets Antimatter as "normal matter moving backwards in time": Time runs left to right in this Feynman diagram of electron-positron annihilation. When interpreted to include retrocausality, the electron (marked e-) was not destroyed, instead becoming the positron (e+) and moving backward in time. Inspired by such an interpretation, "annihilating" a Quantumly Entangled (QE'd) particle could be construed as "sending one of the entangled particles backwards in time", whilst using the resulting antimatter explosion energy to reconstruct "new & freshly laundered matter" (as it were), free from the QE of the previous particle: Merged post follows: Consecutive posts merged YET ANOTHER QUESTION — 'Free Will' is Fore-Ordained ?? Consider "cooking" a proton, to induce Positron Emission: p+ + energy —> n + e+ + ve Then, hold the emitted positron in some sort of magnetic confinement, while you wonder whether to (1) take the positron to the Alpha Centauri star system, and there annihilate it with a local (Alpha Centaurian) electron; or (2) take the positron to the Sirius star system, and there annihilate it with another, also local (Siriusian), electron. From the point of view of 'Free Will', you can choose to do either at your leisure (technology permitting). But, according to Retrocausality, (1) your "positron" is really just an electron moving backwards in time, from the place of its "antimatter annihilation"; and, so, (2) that electron already knows which star system it came from, having already experienced the "bending backwards in time" of its worldline, which you will soon see as the "antimatter explosion" in the ___________ star system, that the electron already knows you chose !! Note that the "conservation of worldlines" requires that the "backwards-in-time moving electron" becomes "buried" in the proton, turning it into a neutron (which is completely consistent w/ observations of (free) neutron decay: n —> p++ e- + energy). (Somewhat similarly, the emitted electron-neutrino ve [not displayed in the above picture] must have been "buried" in the proton all along.) Indeed, in an anthropomorphized sense, the last thing the puzzled 'positron' ponders, as it speeds out of your magnetic cage, and towards the awaiting proton — where, in yet another burst of energy, its worldline will again "bend backwards in time", and start moving "forwards" as a "regular electron" again — where it will soon be "buried" in the new neutron... is why the heck it just heard you wondering where you'd take it, because it already knew where you'd gotten it, all along !!

Quantum Entanglement seemingly requires that the soon-to-be entangled particles must first come into causal contact. Now, ever after, those now-entangled particles may drift away, from the "entanglement event" in Space-Time. Yet, since they travel slower than the speed of light, those now-entangled particles will always remain in the (causal) Future Light-Cone of the "entanglement event" in Space-Time. Thus, the world-lines, of all the now-entangled particles, will always be related causally to their mutual "entanglement event". Has it ever been suggested, that the apparent faster-than-light, "non-local" nature of Quantum Entanglement, can be explained, in a 'Retro-Causal' way, as the flow of information, from the first "entanglement observation event", through the entangled world-lines, to the correlated "entanglement observation event" ? To wit, no information actually travels through the space between the now-entangled particles — instead, the information flow "vibrates" along the now-entangled worldlines ??

Thanks for the reply. The image of the "Buzz Magnets" has become broken, but here's the same (if smaller) picture: Also, Mars' inner moon (Phoebos) is more massive than the outer moon (Deimos)*. Thus, the "mass profile" of Mars' moon system decreases w/ increasing radius. This is conspicuously completely consistent w/ claims, and so seemingly suggests, that "Phoebos & Deimos formed from a debris disk orbiting around Mars, much like the Moon of Earth" (as per PP). * http://en.wikipedia.org/wiki/Deimos_(moon)

Most asteroids are not spherical. For example, the Martian mini-moon Phoebos "is an irregularly shaped rock just less than 28 kilometres across" whose volume is over 15% voids. Of course, the situation is similar for smaller Deimos (Mars' other mini-moon). Now, in addition, according to the National Geographic Channel documentary Naked Science -- Avoiding Armageddon (TV), most asteroids are "rubble piles" of loosely bound bits kept together by gravity. As an analogy, this author reminds the reader about Buzz Magnets, Thus, most asteroids are not solid, but are, rather, pretty porous, w/ potentially vast interstitial spaces. Indeed, when such "rubble piles" pass near enough to the Earth, they suffer "seismic shakes" which turn over the top layer of regolith, exposing fresh material upon the surface*. (This would be analogous to bringing a bunched bunch of Buzz Magnets near enough to a big bar magnet, making them move into a new configuration, shifting & sliding into another shape.) * Comparing the images, of the bunched bunch of Buzz Magnets, against the photo of Phoebos, strongly suggests, that the "rubble pile" comprising the core of an asteroid (= Buzz Magnets) is shrouded in a thick powder of dust (= regolith). So, when such a "rubble pile" passes near enough to Earth (= bringing Buzz Magnets near a big Bar Magnet), as those chunks comprising the core slip & slide into another sort of shape, they naturally churn over the regolith adhering to them. CONCLUSION (??): Non-solidity seems strongly associated w/ non-sphericity. Could it be the case, that -- in the main, as a general rule -- these qualities are actually equivalent ?? To wit, that an asteroid is solid if and only if it is spherical, et vice versa ?? Indeed, when a world becomes big enough for its own self-gravity to shape itself into a sphere, such processes presumably compact & compress all "cave" interstitial spaces. (In analogy, whilst there can be caves on the surfaces of rocky planets, where gravity is weak, their are probably no caverns deep down in the Mantles or Cores of those worlds.) Merged post follows: Consecutive posts mergedADDENDUM: If asteroids are really "rubble piles", perhaps applying "Gravity Tractors" to pull the asteroid off course, or Lasers to vaporize surface rocks to push the asteroid off course, would not work. Instead, perhaps its possible that "pulling" or "pushing" on a "rubble pile" would merely move the bits about, rearranging the rocks into some new shape. Indeed, perhaps the effect would look allot like bringing such "rubble piles" past a planet -- the parts would simply shift & slide, exposing fresh regolith. In-so-far as "Gravity Tractors" & Lasers rely on an assumed solidity of the asteroid, then "rubble piles" would possibly behave in an entirely different manner from such assumptions. Merged post follows: Consecutive posts mergedADDENDUM: Mars' mini-moons (Phoebos & Deimos) have equatorial orbits, much like the Moon of Earth. This seemingly suggests that Phoebos & Deimos formed from a debris disk orbiting around Mars, much like the Moon of Earth. Now, Earth's Moon was made from an impact, w/ a Mars-sized body (10% Mearth), about 4.5 Gya. And, Mars shows signs of a similarly-scaled-sized impact, over 4 Gya, w/ a Moon-sized body (10% Mmars) which bashed out the Borealis Basin covering the northern 40% of Mars' surface. Could it be, that the Phoebos & Deimos mini-moons are really the "rubble-pile" remains, of all the bits & pieces blasted out into orbit by the "Borealis Basin impactor" ?? As with the "Theia impactor" which made the Moon of Earth, the catastrophic collision created a debris disk orbiting about the central planet. But, with Mars, much less material subsequently settled into the debris disk -- so much so, that only two non-solid, non-spherical mini-moons were made*. * According to the National Geographic Channel documentary Traveler's Guide to the Planets -- Mars (TV), tidal friction is pulling Phoebos ever closer to Mars. Phoebos is losing 2m of altitude per century, and is expected to impact Mars in about 50 Myr. Perhaps, then, the Borealis Basin impact event resulted in many mini-Moons... only two of which have survived over 4 Ga of tidal interactions. Indeed, given that planetary formation processes so frequently involve such catastrophic encounters (inner star systems have been described as "shooting galleries"), maybe Moons & mini-moons are quite common across the Cosmos* ?? * According to Star Factories by Ray Jayawardhana, computer simulations suggest that the Theia impact event created one Moon, or two Moons which soon merged (< 50 Myr). Such suggests that, whilst many mini-moons might be quite common, major Moons might most often be single.

2/5 --> 3/5. So, the Earth's S-GBE is actually ~20x the S-BE of TNT, so the threshold mass, where S-GBE = S-BEtnt is closer to one Moon mass. According to Wired Chemist, the Bond Energy of Si-O, a quite common constituent of rocky worlds of terrestrial type, is 452 KJ / mol. Now, a molecule of Si-O masses 44 AMU (roughly), and so a mole of those molecules is 44 g (6e23 x 44 = 2.64e25 AMU = 0.044 kg). So, the "S-BE" of Si-O is ( 452 KJ / mol ) / ( 0.044 kg / mol ) = 1e7 J / kg = 10 GJ / ton, or about 2.5x the S-BE of TNT. So, as Swansont said, the S-BE of TNT is quite characteristic of common chemical bonds, not only of C-H compounds, but even of Silicates as well. Using these revised numbers, the threshold mass (S-GBE = S-BEsi-o) is 3-4 Mmoon. More massive worlds will be "Gravity Bonded", whereas smaller worlds will be "Chemically Bonded". In-so-far as the stronger mutual self-gravity, of rocky material, in "Gravity Bonded" worlds, will translate, ultimately, into some sort of Pressure (as gravity pushes the rocky materials up against each other)... then "Chemically Bonded" planetoids could conceivably behave more like stones on the surface of Earth, at "low pressure"; and, conversely, "Gravity Bonded" planetoids might be more like stones under high pressure. How does pressure affect the property of rocks ?? Is there any evidence of such differences, in bulk properties, between the Earth ("Gravity Bonded") and the Moon ("Chemically Bonded") ?? Extras: According to Wikipedia, 1 eV / AMU = 100 MJ / kg (roughly).

That's an interesting case to consider. Technically, water is an "external object", used to help hunt, so it'd be a "hunting tool". However, the medium that an animal lives in (water, air) seems to be in such intimate & continuous contact w/ the animal, that it doesn't require the same kind of logical leaps to try to use as a tool.

Please ponder a planet's Gravitational Binding Energy (Joules): U = (2/5) G M2 / R Now, please ponder that planet's Specific GBE (J / kg): U = (2/5) G M / R It turns out, that for the Earth, the S-GBE is roughly 15x the Energy-per-Mass released by exploding TNT. Thus, even if the whole world was turned into TNT, and detonated, it wouldn't be blown apart, but would reform, from the fragments, from gravity. Earth is "strongly gravity bound" (as it were). Now, according to Ollivier, Encrenaz, et al, in Planetary Systems — detection, formation, and habitability of extra-solar planets (pp. 209-213), for rocky worlds of terrestrial-type bulk chemical composition, R ~ M0.27. So, as it happens to happen, for a world about twice the mass of the Moon (i.e., M ~= 0.024 Mearth), the S-GBE becomes comparable to the chemical energy stored in TNT. QUESTION: Is the chemical energy, per mass, stored in TNT characteristic of common chemical kinds of bonds ?? And, if so, is there some sort of significance, to this transition, for worlds w/ the Moon's mass, where the S-GBE starts exceeding the characteristic specific energy of chemical bonds ?? I suspect this has some sort of significance, since it marks the transition from "strongly gravity bound" worlds to "weakly gravity bound" worlds, relative to the energy scales associated w/ chemical bonding. So, maybe Moon-mass worlds begin to become "brittle", b/c they're basically bound by bonds of chemical, not gravitational, origin — to wit, maybe Moon-mass worlds are like large single slabs of surface rock, as seen on Earth ("a big ball of crustal rock"). Is there some sort of theorem, that rocks forming under "low pressure" conditions cannot create strong chemical bonds ?? Could GPE be some sort of source for any necessary input energies, for forming rock crystals which are chemically bound ??

Along those lines, I want to offer a definition for "tool for hunting" vs. "weapon for fighting", based upon Animal Psychology. Please ponder the NGC documentary Eternal Enemies (DVD), about the bloody brawls between lions & hyenas, in southern Africa: During the documentary, the hyenas kill the cubs of one lioness, leading to the said cited "revenge" raid. Watching the scenes on screen, it certainly seems like you can viscerally feel the roiling rage of the lions & lionesses against the heckling & harassing hyenas. Now, the Animal Psychology of this rage & hatred has to be different & distinct, from pure predation. Predation, concerned w/ capturing calories, is cold & calculated, and mainly motivated by some sense of starvation ("I'm hungry, so I'll hunt"). But Aggression is concerned w/ inflicting injury, in the heat of passion, and is mainly motivated by anger & rage ("I'm angry, so I'll attack"). Stated in simple speak, these differences -- "cold calculated hunting when hungry" vs. "hot & enraged angry attack" -- could clarify the categorization, of "helpful hunting tools" vs. "wielded weapons". Stated in simple speak, if a tool is used to help in hunting, while hungry, in a cold & calculated way, it's to be treated as a tool. But, if a tool is used, during the peak of passion, having a hot & roiling rage, as an expression of anger, to attack, it's to be treated as a wielded weapon. Animal Psychology, based upon the state of mind of the animal employing the tool, at the time, could categorize & classify the type of tool (help in hunting vs. wielded weapon). To wit, focusing on the "kind of consciousness", of the organism, at the time of tool use, could conceivably classify & categorize the type of that tool -- and, in particular, whether it's a wielded weapon. Merged post follows: Consecutive posts mergedI want to point out, too, that according to the PBS website I cited above, as well as the Discovery Channel documentary Discovering Ardi (DVD), since the Hominin LCA ~7 Mya, along the line leading to modern man, males' conspicuous canine teeth started shrinking from the very first. (Today, modern men have no noticeable canines, compared to the conspicuously fierce fangs of chimps.) Now, amongst most male primates, such conspicuous canines function as fangs for ferocious grimaces of aggression. So, as seen in said cited DC documentary, the remarkable reduction of males' canines probably implies that bitings became an increasingly infrequent & unimportant part of male aggression. And, this is completely consistent, w/ claims that clubs, as wielded weapons, were an increasingly important part of male aggression amongst early "proto-human" Hominins. (In essence, male "proto-humans" outsourced their aggression, from teeth to tools, the former dramatically decreasing in size, whilst the latter inexorably increased in sophistication.) If so, this would imply that human evolution has been profoundly influenced by tools & technologies, for roughly 7 Myr.

Chimpanzees brandish branches as clubs, both as part of display, and -- if tensions escalate -- during aggressive acts against other Chimps. According to said cited show (as per PP), Bonobo males begin brandishing branches as clubs, during displays... until the females assert their authority, the males "give up", and quiet down. The impression is given, that the only reason Bonobo males don't actually carry clubs into combat, is b/c the females block the behavior -- left to their own devices, Bonobo males act essentially the same as Chimps. Such apparently indicates the important impact of culture on primate behavior. Merged post follows: Consecutive posts mergedI'd like to clarify, and ask again, if any other Earth animals "wield weapons" -- for example, do elephants pick up logs, in the trunks, to throw at lions, or, do birds pick up rocks, in their talons, to drop on potential predators ?? Do Galapagos Finches "joust" against each other, w/ cactus spines ?? (Thanks again in advance for any info.) Merged post follows: Consecutive posts mergedTerrain Use as Tool Use According to the NGC documentary Relentless Enemies (DVD;book), lionesses will herd wildebeests past particular types of terrain, which help the hunters, by providing positions for ambushes, by other members of the pride. This amounts to "using external objects" to help in hunting, and so is somewhat similar to Monkeys & Otters using stones ("external objects") to help in acquiring calories. Now, (some) Chimps go one better, and actually shape & sharpen their spear-skewers. This would be analogous to lionesses (say) artificially modifying terrain, to help in hunting, by building blinds from logs & other lumber lying around (say). Are there any other (Earth) species known to modify terrain to help in hunting (creating camouflage, constructing traps, etc.) ??

What about whales being marine mammals, whereas sauropods were terrestrial animals ? Would that make any difference ?

According to Science Illustrated [Nov./Dec. 2008 AD], pg. 33: So, sleep cycles are ubiquitous to all Earth animals. And, all smarter & more neurologically advanced species need more sleep. And, all Herbivores have evolved to sleep less, surely b/c: Herbivorous diets require far more time to consume Herbivores, consuming less protein, typically possess smaller brains So, surely, this is why the huge dinosaurian Sauropods had such small, practically pea-sized brains — they were so big, they literally could never stop to sleep, their round-the-clock gorging requiring them to simplify their brains & neurology, as much as possible, so they could keep crashing through the forests, all night, and all day. Primitive mammals must have slept each night, deep in their dens, to the deep foot-thumps and bellows of these big beasts ! Possible ? Plausible ??

A stronger circumstantial case can be built about use of (simple) "sticks & stones". For, all Hominins (humans, chimps & bonobos) wield, as weapons, such simple "sticks & stones". According to the PBS documentary The Last Great Ape (DVD), when bonobo males meet another bonobo clan, they initially react allot like common chimps — grabbing & shaking tree branches, and picking up sticks to brandish as clubs. And, I understand, that all Hominins chuck rocks at potential predators. Thus, the basic proposition, that the LCA wielded weapons, such as (simple) sticks & stones, surely seems to stand. But it is somewhat less likely that the LCA crafted & fashioned weapons, such as shaped & sharpened spears & skeweres. Such could quite conceivably represent independent developments, during the last ~7 Myr. Merged post follows: Consecutive posts merged Is this a native, wild behavior ? Like Otters cracking open seashells w/ rocks, or like Monkeys cracking open nuts w/ rocks, this represents "hunting" — using a tool to capture calories for food. But, are there any Terrestrial animals, besides Hominins' simple sticks & stones, known to wield "weapons" — tools used to cause death & wounds in warfare ?? (Thanks in advance for any information.)

Thanks for the link

Chimpanzees use stone tools, and fashion sharpened spears (by chewing). Chimps & Humans diverged, evolutionarily, about 6 million years ago*. * http://en.wikipedia.org/wiki/Chimpanzee ; http://www.buzzle.com/articles/chimpanzee-habitat.html This strongly suggests, that the Last Common Ancestor (LCA) of Chimps & Humans made & used stone tools & spears, before about 6 million years ago. So, seemingly, various creatures have been crafting increasingly sophisticated weapons systems for many millions of years. The spear, specifically, may be much more ancient than typically proclaimed. Possible ? Plausible ?

According to Wikipedia, the Raleigh Convection Index, for the onset of natural convection in systems heated from below (as w/ air heated from below by burning materials), due to Thermal Expansion, is: [math]Ra = \frac{\alpha}{\kappa \, \mu} \rho g \Delta T L^{3} = \frac{k}{\rho \, c_{p}} \frac{1}{\kappa \, \mu} \rho g \Delta T L^{3} = \frac{k \, \Delta T}{c_{p} \, \kappa \, \mu} \times g L^{3}[/math] Now, for "forest fires" on an hypothetical exoplanet, would not the characteristic convection length scale (L) basically be that world's Atmospheric Scale Height (Wiki.) ?? [math]H = \frac{k_{B} \, T}{\bar{m} \, g}[/math] If so, an exoplanet's convection score for "forest fires" would be: [math]Ra = \left( \frac{k \, \Delta T}{c_{p} \, \kappa \, \mu} \right) \left( \frac{k_{B} \, T}{\bar{m}} \right) \times g^{-2}[/math] Isolating the effects of the exoplanet's Surface Gravity (essentially assuming similar atmospheric conditions & chemical compositions), Ra decreases as g-2. QUESTION: Is it realistic, then, for Science Fiction (say), to claim that conflagrations are somewhat suppressed upon bigger planets, but increasingly enhanced, on smaller worlds ?? To wit, that fires affecting an outpost (say) upon a Terraformed Mars (g = 0.4) might be half-a-dozen times more prone to forming full-fledged firestorms, as opposed to those upon this particular planet ??

From their figure 21.16, Specific Angular Momentum is the ratio of Angular Momentum to Mass, L / M. Since (ignoring coefficients) L = I w, and I = M R2, SAM has the units of R2 w (as you said).

Minimum Mass Solar Nebula (MMSN) model Caleb A. Scharf (Extrasolar Planets & Astrobiology, pg. 101) mentions the MMSN model, for the (vertically integrated) surface density, of the proto-Solar proto-Planetary disk: [math]\sigma® \equiv \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math] where [math]\sigma_{0} = 1700 \; g \; cm^{-2}[/math] and [math]r_{0} = 1 AU[/math]. The total disk mass is estimated to have been about [math]M_{disk} \approx 0.018 \; M_{\odot} \approx 6000 \; M_{\oplus}[/math]. Now, for the general case, of extra-Solar proto-Planetary disks, we have, for the disks' masses & angular momentums, the following mathematical definitions: [math]M_{disk} = \int 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math] [math] = 2 \pi \sigma_{0} r_{0}^{2} \int dx \; x^{-1/2}[/math] [math] \equiv 2 \pi \sigma_{0} r_{0}^{2} \; I_{1}[/math] and: [math]L_{disk} = \int 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} \sqrt{G \; M_{*} \; r}[/math] [math] = 2 \pi \sigma_{0} r_{0}^{2} \sqrt{G \; M_{*} \; r_{0}} \int dx [/math] [math] \equiv \frac{M_{disk}}{I_{1}} \sqrt{G \; M_{*} \; r_{0}} \; I_{2}[/math] Knowing the total disk mass (above), and angular momentum (Carroll & Ostlie. Intro. to Mod. Astrophys., pg. 893), we can calibrate these equations. We find, w/ approximate self consistency, that the best fit values for the integrals are: [math]I_{1} \approx 15[/math] [math]I_{2} \approx 3[/math] Now, we know (Carroll & Ostlie, ibid.) that for stars [math]\leq 2 M_{\odot}[/math]: [math]\frac{L_{tot}}{M} \propto M^{2/3}[/math] [math]\frac{L_{*}}{M} \propto M^{16/3}[/math] The residual difference between these values has probably been deposited into (inferred) Planetary systems (Carroll & Ostlie, ibid.). Therefore, we define [math]L_{disk} \equiv L_{tot} - L_{*}[/math]. Furthermore, we know that the Planetary disk mass is zero for [math]2 M_{\odot}[/math] stars, and [math]0.018 M_{\odot}[/math] for [math]1 M_{\odot}[/math] stars (e.g. Sun). In addition, we doubt that the Planetary disks of minimum-mass stars [math]\left( M \approx 0.08 M_{\odot} \approx 27,000 M_{\oplus} \right)[/math] can exceed the mass of those central stars. With these two data points, plus the (presumed) upper bound, we can confidently allege a linear increase of disk mass, with decreasing stellar mass, as: [math]M_{disk} \approx 6000 M_{\oplus} \times \left( 2 - \frac{M_{*}}{M_{\odot}} \right)[/math] Having an empirical formula for disk angular momentum [math]\left( L_{disk}\right)[/math], and a plausible estimate for disk mass [math]\left( M_{disk} \right)[/math], as functions of stellar mass [math]\left( M_{*} \right)[/math], we can, in principal, use the aforementioned mathematical definitions (along w/ their best fit values for [math]I_{1}[/math] and [math]I_{2}[/math]) to deduce the best fit values for the parameters [math]\sigma_{0}[/math] ("central" density) and [math]r_{0}[/math] (radial scale length), as: [math]r_{0} \equiv \frac{1}{G M_{*}} \left( \frac{L_{disk}}{M_{disk}} \frac{I_{1}}{I_{2}} \right)^{2}[/math] and: [math]\sigma_{0} \equiv \frac{M_{disk}}{2 \pi I_{1}} \frac{1}{r_{0}^{2}} = \frac{\left( G M_{*}\right)^{2}}{2 \pi} \left( \frac{M_{disk}}{I_{1}}\right)^{5} \left( \frac{L_{disk}}{I_{2}}\right)^{-4}[/math] Units can be verified by inspection. SciLab was used to numerically calculate the Disk Scale Radii & Disk Central Densities, as functions of stellar mass: Fig. 1 -- Disk Scale Radius vs. Stellar Mass Fig. 2 -- Disk Central Density vs. Stellar Mass It was numerically verified, that these values do, indeed, reproduce the original estimate for total disk mass (above). Now, the MMSN model is defined as having [math]Z = 0.01[/math] abundance of rocky materials (Caleb A. Scharf, ibid.). For our Solar System, then, we may define [math]\Delta[/math] to be the (relative) distance, to either side of the Earth's orbit, which must be swept up to accumulate [math]1 M_{\oplus}[/math] of rocky material: [math]1 M_{\oplus} = Z \int_{1 AU \times (1-\Delta)}^{1 AU \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} = Z \left( 2 \pi \sigma_{0} r_{0}^{2} \right) \int_{1-\Delta}^{1+\Delta} dx \; x^{-1/2}[/math] Substituting from the formula for total disk mass (above), we have that: [math]1 M_{\oplus} = Z \frac{M_{disk,\odot}}{I_{1}} \int_{1-\Delta}^{1+\Delta} dx \; x^{-1/2}[/math] The solution is straightforward, and yields [math]\Delta \approx 1/8[/math]. Armed with this definition for [math]\Delta[/math], we can calculate how much mass lies within the equivalent regions, of the Habitable Zones, around other stars. The only additional information required is the evolution, of the distance to said Habitable Zones, as a function of stellar mass. Now, the definition of the Habitable Zone is that: [math]\frac{L_{*}}{D_{HZ}^{2}} = constant[/math] And, to close approximation (Bowers & Deeming. Stars, pg. 28), [math]L_{*} \propto M_{*}^{4}[/math]. So, [math]D_{HZ} \propto M_{*}^{2}[/math]. Therefore, following the foregoing, we define: [math]M_{HZ,rock} \equiv Z \int_{D_{HZ} \times (1-\Delta)}^{D_{HZ} \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2}[/math] [math] = Z \frac{M_{disk}}{I_{1}} \int_{\frac{D_{HZ}}{r_{0}} \left( 1-\Delta \right) }^{\frac{D_{HZ}}{r_{0}} \left( 1+\Delta \right) } dx \; x^{-1/2}[/math] [math]= Z \frac{M_{disk}}{I_{1}} \sqrt{\frac{D_{HZ}}{r_{0}}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}[/math] [math]= 1 M_{\oplus} \times \frac{M_{disk}}{M_{disk,\odot}} \times \sqrt{\frac{D_{HZ}}{r_{0}}} [/math] This solution is also straightforward, and SciLab was used again, to numerically calculate these estimated Habitable Zone masses of rocky material, as a function of stellar mass: Fig. 3 -- Habitable Zone Rock Mass vs. Stellar Mass Apparently plausibly, the amount of rocky mass, in the central star's Habitable Zone, increases inexorably, with decreasing stellar mass [math]\leq 2 M_{\odot}[/math]. Finally, we can calculate quantities associated with the Angular Momentum, of that rocky mass, inside said stars' Habitable Zones. First, we can define the total Angular Momentum, of all that rock, revolving around in the disk: [math]L_{HZ,rock} \equiv Z \int_{D_{HZ} \times (1-\Delta)}^{D_{HZ} \times (1+\Delta)} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} \sqrt{G M_{*} r}[/math] [math] = Z \frac{M_{disk}}{I_{1}} \sqrt{G M_{*} r_{0}} \int_{\frac{D_{HZ}}{r_{0}} \left( 1-\Delta \right) }^{\frac{D_{HZ}}{r_{0}} \left( 1+\Delta \right) } dx[/math] [math] = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy[/math] Next, we can define the "Keplerian" Angular Momentum, of all that rock, once collapsed into a single Planetesimal, orbiting at that star system's scaled equivalent of 1 AU: [math]L_{Kepler} \equiv M_{HZ,rock} \times \sqrt{G M_{*} D_{HZ}}[/math] [math]= Z \frac{M_{disk}}{I_{1}} \sqrt{\frac{D_{HZ}}{r_{0}}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2} \times \sqrt{G M_{*} D_{HZ}}[/math] [math] = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}[/math] Now, presumably, any excess of [math]L_{HZ,rock}[/math] over [math]L_{Keplerian}[/math] will set the Planetesimal spinning on its axis. We therefore define [math]L_{spin} \equiv L_{HZ,rock} - L_{Kepler}[/math], which we divide by the Planetesimal's mass [math]M_{HZ,rock}[/math] as per normal normalization. Noting that: [math]M_{HZ,rock} = Z \frac{L_{disk}}{I_{2}} \frac{D_{HZ}}{r_{0}} \int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2} \times \frac{1}{\sqrt{G M_{*} D_{HZ}}}[/math] this yields: [math]\left( \frac{L}{M} \right)_{planet} = \sqrt{G M_{*} D_{HZ}} \times \left( \frac{\int_{1-\Delta}^{1+\Delta} dy}{\int_{1-\Delta}^{1+\Delta} dy \; y^{-1/2}} -1 \right)[/math] [math]= \sqrt{G M_{*} D_{HZ}} \times \left( \frac{\Delta}{\sqrt{1+\Delta} - \sqrt{1-\Delta} } -1 \right)[/math] [math]\approx \sqrt{G M_{*} D_{HZ}} \times \left( 0.998 - 1 \right)[/math] [math]\approx -0.002 \sqrt{G M_{*} D_{HZ}} [/math] [math]\propto - M_{*}^{3/2}[/math] Assuming planets typically rotate prograde, this analysis seemingly suggests that planets preferentially accumulate material from beyond their orbit [math]\left( r > r_{p} \right)[/math], rather than from within it [math]\left( r < r_{p} \right)[/math]. It is remotely possible, that these results imply, that bigger & brighter stars probably possess planets that tend to spin (rotate) more rapidly. Lastly, note that the change in Specific Angular Momentum (SAM), across the star's Habitable Zone, increases strongly with stellar mass: [math]\delta \left( \frac{L}{M} \right) = \delta \sqrt{G M_{*} D_{HZ} } = \frac{1}{2} \sqrt{\frac{G M_{*}}{D_{HZ}}} \delta D_{HZ} = \sqrt{G M_{*} D_{HZ} } \times \frac{\Delta}{2} [/math] [math]\propto M_{*}^{3/2}[/math] Thus, in-so-far as SAM poses the biggest obstacle to the accretion of Planetesimals, smaller & dimmer stars may be more likely to have fewer but (much) bigger planets, whereas bigger & brighter stars might be more likely to have more but (much) smaller planets. For example, an M-Class red dwarf might have one Super-Earth in its inner system, and one or two Super-Jupiters in its outer system. Conversely, an F-Class green dwarf might have many Mars- & Mercury-sized planets in its inner system, and many Uranus- & Neptune-sized gas giants in its outer system. Merged post follows: Consecutive posts mergedFigure 21.16 from Carroll & Ostlie (ibid.) http://www.freeimagehosting.net/image.php?fbaed847c5.jpg Merged post follows: Consecutive posts mergedDisk Zone Masses It is straightforward to define Disk Zone Masses: [math]M_{zone} = \int_{r_{i}}^{r_{o}} 2 \pi r dr \; \sigma_{0} \left( \frac{r}{r_{0}} \right)^{-3/2} = 2 \pi \sigma_{0} r_{0}^{2} \int_{x_{i}}^{x_{o}} dx \; x^{-1/2}[/math] [math][/math] [math] = \frac{M_{disk}}{I_{1}} \int_{x_{i}}^{x_{o}} dx \; x^{-1/2} [/math] Now, stellar proto-Planetary Disks are divided into four successive zones (Carroll & Ostlie, ibid., pg. ~898): Metal Zone -- only metals solidify Rock Zone -- metals & rocks solidify Ice Zone -- beyond Snow Line, water ice also condenses Methane Ice Zone -- beyond Methane Snow Line, methane ices also condense From the information of our Solar System, the "Metal Line" (0.5 AU), Snow Line (5 AU), and Methane Snow Line (30 AU) can be scaled for other star systems, precisely as per [math]D_{HZ}[/math] above. SciLab was used to numerically calculate these quantities, and plot said Zone Masses, as functions of stellar mass: Fig. 4 -- Zone Masses vs. Stellar Mass It is found, that a star-system-wide scaling factor floss ~ 0.10 cleanly & consistently accounts for the observed amounts of mass remaining in our Solar System. (For example, out of the inferred initial Solar proto-Planetary Disk mass of [math]6000 M_{\oplus}[/math], over [math]20 M_{\oplus}[/math] of rock material remained, according to calculations, in the Rock Zone, while over [math]4000 M_{\oplus}[/math] is calculated to have resided in the Ice Zone (Jupiter, Saturn, Uranus), and over [math]200 M_{\oplus}[/math] is calculated to have resided in the Methane Ice Zone (Neptune+). Across the Solar System, all these values strongly suggest a "retention factor" of roughly 10%.) These reduced "residual masses", for the Ice- & Methane-Ice Zones, appear in the above plot, alongside the unscaled Rock Zone masses (effectively x10) for purposes of visualization. It is immediately apparent, that, b/c of the inexorably increasing comparative coreward concentration of disk mass, with decreasing star mass, the most massive planet-bearing stars (F- & A-Class) actually boast the biggest Outer Star Systems (Ice- & Methane-Ice-Zones). In particular, the A-Class stars [math]\left( M > 1.5 M_{\odot} \right)[/math] boast the biggest Neptune-like Ice Giants, some surely exceeding [math]100 M_{\oplus}[/math]. This is (practically) completely consistent, with the observations from Fomalhaut System, which boasts a (single) big Ice Giant, (Fomalhaut b) (~115 AU) and Kuiper Belt (~150 AU), out in its Methane Ice Zone (beyond ~125 AU). Conversely, the smallest stars support the most massive Inner Star Systems, housing [math]5-8 M_{\oplus}[/math] of rocky planetary material. However, with so much mass, crammed into such confined regions, it surely seems somewhat certain that such planetary systems could become incredibly unstable, and "torque themselves to shreds", perhaps explaining why small stars seemingly mostly lack Planetary Systems: Fig. 5 -- Hertzsprung-Russell Diagram in Luminosity - Mass space, also indicating known Exoplanet-bearing systems (W.T. Sullivan III & J.A. Baross., Planets & Life, pg. 445; cf. link). Merged post follows: Consecutive posts mergedEstimating Number & Mass of HZ Exoplanets Consider two Planetesimals , each of mass [math]m[/math], orbiting their central star. Their (assumed circular) orbits are separated by a distance [math]\delta D[/math]. From Kepler's Laws, we know that the difference in Angular Momentum, between both orbits, is: [math]\Delta L = m \sqrt{G M_{*}} \frac{\delta D}{2 sqrt{D}}[/math] Now, for both Planetesimals to merge, they must match their Angular Momentums, which requires Torques. We (crudely) guestimate the magnitude of said torques, with a simple model, of periodic pulses of strong gravitational influence, at every planetary alignment: [math]\tau = F \times \Delta D \approx \frac{G m^{2}}{\Delta D^{2}} \times \Delta D[/math] [math][/math] [math]\Delta t \propto \Delta \omega^{-1} = \left( \frac{3}{2} \sqrt{\frac{G M_{*}}{D^{5}}} \Delta D \right)^{-1}[/math] Setting [math]\Delta L \equiv \tau \times \Delta t[/math], we have, after rearranging: [math]\left( \frac{\Delta D}{D} \right)^{3} \propto \frac{4}{3} \left( \frac{m}{M_{*}} \right)[/math] [math][/math] [math]\frac{\Delta D}{D} \propto \left( \frac{m}{M_{*}} \right)^{1/3}[/math] Strictly speaking, the characteristic Planetesimal mass [math]m[/math] is most probably proportional to the retained Rock Zone mass [math]\left( M_{rock} \times Z \times f_{loss} \right)[/math]. But since most material resides in the Rock Zone (i.e., Inner Star System), we can closely approximate the right result w/ [math]M_{rock} \approx M_{disk}[/math]. Then we have that: [math]\frac{m}{M_{*}} \approx Z \; f_{loss} \frac{M_{disk}}{M{*}} = Z \; f_{loss} \frac{0.018 M_{\odot}}{M_{*}} \left(2 - \frac{M_{*}}{M_{\odot}} \right)[/math] [math][/math] [math] = Z \; f_{loss} \; 0.018 \times \left( \frac{2}{\mu} - 1 \right) [/math] where [math]\mu \equiv M_{*} / M_{\odot}[/math]. SciLab was used to calculate the (slightly) more accurate estimates, as originally defined: Fig. 6 -- Estimated Number & Mass of HZ Exoplanets vs. Star Mass For this figure, it was assumed that our Solar System supports 2.5 Habitable Zone (HZ) planets (Venus, Earth, Mars), so that a relative [math]\Delta D / D = 2.5[/math] 'should' sweep out its entire HZ, amassing all the (retained rocky) material, into a single planet. Thus, the estimated number of HZ planets is merely the inverse of the relative [math]\Delta D / D[/math], and the estimated mass per HZ planet is merely the (retained rocky) material mass divided by that above value. This analysis seemingly strongly suggests, that bigger brighter stars support many more Mars-, Mercury-, and Moon-mass worlds (scattered across their vast HZs), whereas smaller stars support single Super-Earths.

Fig. 21.16, on page 893, of Carroll & Ostlie's Intro. to Mod. Astrophys. [1st ed.], shows that (1) for stars more massive than [math]2 M_{\odot}[/math] (A5), Specific Angular Momentum (L/M) increases as M2/3; and, that (2) for stars less massive than said same value, the star's S.A.M. increases as M5 (with the deficit probably borne by planets, according to the caption). We seek an order-of-magnitude calculation, which can indicate some sort of suggestion, as to the origin of this [math]2 M_{\odot}[/math] threshold. S.A.M. for massive stars Please ponder an idealized Giant Molecular Cloud core, of constant density, from which a spherical fragment begins to collapse. The initial Angular Momentum of said spherical cloud at onset of collapse is: [math]L = \frac{2}{5} M R^{2} \omega_{g} = \frac{2}{5} M \left(\frac{M}{\frac{4 \pi \rho}{3}}\right)^{2/3} \omega_{g}[/math] And so: [math]\frac{L}{M} = \frac{2}{5} \left(\frac{M}{\frac{4 \pi \rho}{3}}\right)^{2/3} \omega_{g} \propto M^{2/3}[/math] Thus, it is completely consistent, w/ Carroll & Ostlie (ibid.), to consider the collapse of cloud cores as occurring in a constant density, and w/ a constant initial angular velocity (being [math]\omega_{g}[/math], the local galactic angular velocity, of rotation around the galactic center). More massive stars have higher S.A.M. merely b/c they "reach out farther" across the (quasi-)constant density cloud core, accumulating amounts of matter (dM) having higher & higher S.A.M. (dL ~ dM x R2). Note that, at the threshold mass ([math]2 M_{\odot}[/math]), the S.A.M. is 1017.3 cm2 s-1, from which we can calculate the (effective) cloud core density, as: [math]\rho = \left( \frac{2 \omega_{g}}{5 \frac{L}{M} } \right)^{3/2} M \frac{3}{4 \pi} \approx 7.3 \times 10^{-17} g \; cm^{-3}[/math] Reassuringly, this value favorably compares w/ Carroll & Ostlie's figures for dense cloud cores (~10-16). But, below the threshold mass value of [math]2 M_{\odot}[/math], more and more of that S.A.M. is missing from the central star (& seemingly deposited into a disk of protoplanets). Why would this be? Merged post follows: Consecutive posts mergedThis author's investigation of Extra-solar Planetary Disks: http://www.scienceforums.net/forum/showthread.php?t=47222